OGLE-2014-BLG-0962 and a Comparison of Galactic Model Priors to Microlensing Data

The OGLE observations are conducted with the 1.3 m Warsaw telescope at the Las Campanas Observatory in Chile, with a 1.4 deg² field of view (FOV) camera. Data in the I band were taken at a nominal cadence of Γ = 1 hr⁻¹. The regions around and between the caustic crossings are well sampled. V-band data were also acquired at a lower cadence. Four points, including one near peak, were captured in the OGLE V band between HJD' = 6810 and 6830. OGLE photometry was reduced with the difference-imaging analysis (DIA) method (Alard & Lupton 1998; Wozniak 2000).

The Microlensing Observations in Astrophysics (MOA) collaboration also observed this event (MOA-2014-BLG-285). MOA provided an alert on 2014 May 31. The MOA survey is conducted from the University of Canterbury Mt. John Observatory in New Zealand, which features a 1.8 m telescope with a camera whose FOV is 2.2 deg². MOA observes in the custom R_MOA band, which is approximately the superposition of the standard I and R bands. Normally MOA surveys at high cadence, though for this particular event it missed the portion of the light curve between the caustic entrance and exit because of weather. MOA photometry is reduced by the DIA pipeline summarized in Bond et al. (2001).

The Wise microlensing survey (Shvartzvald et al. 2016b) used the 1 m telescope at Tel Aviv University's Wise Observatory in Israel. The Large Area Imager for the Wise Observatory camera (with FOV = 1 deg²) was used to collect data on this event in survey mode. Wise observations are conducted in the I filter. The nominal cadence is 30 minutes, averaging five to six observations per night because of target visibility. Photometry for Wise is performed with the DIA software described in Albrow et al. (2009).

OB140962 was observed in the first season of the Spitzer microlensing campaign, before the development of the objective target selection procedure of Yee et al. (2015a), which was used in subsequent seasons. Spitzer observations were taken in the L band (3.6 μm). The target was selected for Spitzer observations before it showed any features attributable to a binary. Observations began on 2014 June 6 (HJD' = 6814.59). The last data point was taken on 2014 July 10 (HJD' = 6849.34). A total of 31 observations were taken at a cadence of Γ ∼ 1 day⁻¹. It captured several points during the anomaly. Spitzer photometry was reduced with the pipeline presented in Calchi Novati et al. (2015b).

Purely photometric (i.e., Poisson) errors are often underestimated, which prompts a rescaling of the data points and renormalization of the errors. Except for the OGLE errors, which are rescaled based on the recommended procedure of Skowron et al. (2016), errors from the other observatories (in magnitudes) are modified using the scheme of Yee et al. (2012),²⁷ where

$$\begin{eqnarray}&&{\sigma }_{\mathrm{rescaled}}=k{\sigma }_{\mathrm{pipeline}}.\end{eqnarray} \tag{ 1 }$$

With σ_rescaled, every data point should contribute ∼1 in χ² on average. The k factor is adjusted manually until this is the case. The newly scaled errors are used to reject outliers and updated iteratively. As a result, we reject four OGLE, two MOA, and two Wise measurements. The final adopted k parameters are listed in Table 1.

Six fundamental parameters are associated with and routinely measured for binary lensing light curves: (t₀, u₀, t_E, s, q, α). The first three quantities stand for the peak time, impact parameter of the source trajectory to the lens (scaled to the lens Einstein radius, θ_E; see below), and the Einstein timescale, that is, the characteristic width of the portion of the light curve undergoing magnification. The second set of three parameters pertains to the binary lens. They are the instantaneous projected separation between the components (normalized to θ_E), their mass ratio, and the projected angle of the source trajectory to the binary axis, respectively.

All of the parameters presented so far are either geometric or relative. Of course, it is the absolute physical properties of the lens that are of greatest interest. The lens mass (M_L), distance (D_L), and relative proper motion between the lens and source (μ_rel) can be determined provided additional effects are measured. They are linked to the direct observables via the Einstein radius (θ_E) and the dimensionless vector microlensing parallax (${{\boldsymbol{\pi }}}_{{\rm{E}}}$). For ${\pi }_{\mathrm{rel}}\equiv {\pi }_{L}-{\pi }_{S}=\mathrm{au}({D}_{L}^{-1}-{D}_{S}^{-1})$, D_S being the source distance (usually close to the Galactic center at ∼8.3 kpc), θ_E and ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ are defined as follows (e.g., Gould 2000):

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}\equiv \sqrt{\kappa {M}_{L}{\pi }_{\mathrm{rel}}};\,\,\,\,\,\kappa \equiv \displaystyle \frac{4G}{{c}^{2}\mathrm{au}}\approx 8.14\,\mathrm{mas}\,{M}_{\odot }^{-1};\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&{{\boldsymbol{\pi }}}_{{\rm{E}}}\equiv \displaystyle \frac{{\pi }_{\mathrm{rel}}}{{\theta }_{{\rm{E}}}}\displaystyle \frac{{{\boldsymbol{\mu }}}_{\mathrm{rel}}}{| {\mu }_{\mathrm{rel}}| };\,\,\,\,\,{\mu }_{\mathrm{rel}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{{t}_{{\rm{E}}}}.\end{eqnarray} \tag{ 3 }$$

Manipulating Equations (2) and (3), we find that both the lens mass and distance can be expressed as a function of θ_E and π_E:

$$\begin{eqnarray}&&{M}_{L}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{\kappa {\pi }_{{\rm{E}}}};\,\,\,\,\,{D}_{L}=\displaystyle \frac{\mathrm{au}}{{\pi }_{{\rm{E}}}{\theta }_{{\rm{E}}}+\mathrm{au}/{D}_{S}}.\end{eqnarray} \tag{ 4 }$$

For caustic crossing events, the finite source effect constrains a seventh parameter ρ, where

$$\begin{eqnarray}&&\rho \equiv \displaystyle \frac{{\theta }_{\star }}{{\theta }_{{\rm{E}}}}=\displaystyle \frac{{t}_{\star }}{{t}_{{\rm{E}}}},\end{eqnarray} \tag{ 5 }$$

that is, the size of the source θ_⋆ measured in units of θ_E. If the source radius can be deduced independently, for instance from the event's position in the local color–magnitude diagram (CMD), then θ_E can be calculated. Alternatively, ρ can be defined as the source self-crossing time, t_*, relative to t_E.

The microlensing parallax, ${{\boldsymbol{\pi }}}_{{\rm{E}}}$, can be measured with a second line of sight to the event, which generally will result in a light curve with timing and morphology that are distinct from the first because of the apparent difference in trajectory. A useful qualitative approximation for the components of ${{\boldsymbol{\pi }}}_{{\rm{E}}}$ is given by the scaled difference in t₀ and u₀ between the two sight lines (e.g., Refsdal 1966):

$$\begin{eqnarray}&&{{\boldsymbol{\pi }}}_{{\rm{E}}}=\displaystyle \frac{\mathrm{au}}{{D}_{\perp }}\left(\displaystyle \frac{{\rm{\Delta }}{t}_{0}}{{t}_{{\rm{E}}}};{\rm{\Delta }}{u}_{0}\right)\end{eqnarray} \tag{ 6 }$$

where ${{\boldsymbol{D}}}_{\perp }$ is the projected separation vector between the two observing locations in the plane of the sky. Equation (6) applies for the coordinate system in which the x axis is aligned with ${{\boldsymbol{D}}}_{\perp }$. For Earth and the Spitzer satellite, the magnitude of this vector is approximately 1–1.5 au. Refer to, for example, Equations (8) to (10) in Calchi Novati et al. (2015a) for the exact relation between ${{\boldsymbol{\pi }}}_{{\rm{E}}}$, Δt₀, Δu₀, and instantaneous ${{\boldsymbol{D}}}_{\perp }$, which is used in the actual modeling for ${{\boldsymbol{\pi }}}_{{\rm{E}}}$.

The ground-based light curve has a broad double-horned structure characteristic of roughly equal mass-ratio binary lensing, exhibiting a clear caustic entrance (HJD' = HJD − 2450000 ∼ 6816.6) and exit (HJD' ∼ 6818.3). The hump in the light curve immediately following the caustic exit (HJD' ∼ 6819.0) implies a cusp approach. The shape and timing of these features place tight constraints on the event geometry. Below we illustrate the back-of-the-envelope process of converting the light curve components into microlensing parameters and interpret the inferred physical properties of the source–lens system.

We approximate this event to have zero blending and estimate the Einstein timescale (t_E) from the half-width of the magnified portion of the light curve at 1.3× the baseline flux. For this event, t_E ∼ 5 days. The caustic crossings resolve the source size (i.e., finite source effect), allowing us to determine θ_E from Equation (5). Figure 1 shows the half-width of the caustic entrance is t_⋆ ∼ 0.15 day, so ρ ≈ 0.03. To obtain θ_⋆, we note that the event placement in the local CMD is consistent with a bulge giant (Figure 2 and Section 4.1). A typical clump giant might have a radius 5–10× that of the Sun, say θ_⋆ ≈ 4 μas. Then, θ_E ≈ 0.13 mas. Together with t_E and Equation (3), we find that the relative lens–source proper motion is μ_rel ≈ 9.5 mas yr⁻¹.

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For many microlensing discoveries with only ground-based data, deducing θ_E and μ_rel is as far as we can go. To estimate the absolute physical properties of the lens, we might assume the lens has a typical distance of D_L ∼ 6 kpc. Assuming a source distance of D_S ∼ 8.3 kpc, π_rel ≈ 0.046. Then, substituting θ_E ≈ 0.13 mas into Equation (2) gives M_tot ≈ 0.045 M_☉. Therefore, the atypically small θ_E for this event (normally ∼0.5 mas) implies an exciting low-mass brown dwarf (BD)–BD binary.

For this event we have parallax information, which constrains the true lens distance and mass. To estimate π_E, we see from Equation (6) that $| {\pi }_{{\rm{E}}}| \approx \sqrt{{({\rm{\Delta }}{t}_{0}/{t}_{{\rm{E}}})}^{2}+{({\rm{\Delta }}{u}_{0})}^{2}}$, where we have used the fact that D_⊥ is of order 1 au. For OB140962, the Spitzer light curve actually closely mimics the ground-based one in shape as well as timing, but offset by Δt₀ ∼ 0.3 day. The virtually indistinguishable light curve morphologies between the two sight lines strongly imply nearly identical impact parameters between the two events (i.e., Δu₀ is negligible), so π_E ∼ Δt₀/t_E ≈ 0.06. The physical parameters M_tot and D_L can both be computed from θ_E and π_E. According to Equation (4), M_tot ≈ 0.27 M_☉ and D_L ≈ 7.8 kpc. Here we have again assumed that the source is located at the Galactocentric distance R₀ ∼ 8.3 kpc. Therefore, from this heuristic evaluation of the ground light curves in conjunction with the Spitzer parallax, we reach the conclusion that the lens is a typical low-mass binary that must be very close to the source. This is at odds with the earlier expectation of a very low mass lens from θ_E and μ_rel alone.

To map the overall topology of the parameter space for the ground-based light curve, we perform a grid search in χ² space over $\mathrm{log}s$, $\mathrm{log}q$, and α, which are responsible for the magnification profile of the event (Dong et al. 2006). For each point $(\mathrm{log}s\in [-1,1],\mathrm{log}q\in [-5,1],\alpha \in [0,2\pi ])$ on the 100 × 100 × 21 grid, we allow the other light curve parameters (t₀, u₀, t_E, ρ) to be explored by a Markov chain Monte Carlo (MCMC) algorithm until it settles on the minimal χ². We find the global minimum χ² to be in the region around $\mathrm{log}s\sim 0.3$ and $\mathrm{log}q\sim 0.07$ (based on modeling the OGLE I-band data prior to error rescaling).

From the best-fit grid point, we launch our full MCMC for a joint fit for all four data sets (three ground, one space). The finite source effect is modeled using the ray-shooting method (Kayser et al. 1986; Schneider & Weiss 1986; Wambsganss 1997), and regions of the light curve immediately adjacent to caustic crossings are computed through the hexadecapole approximation (Gould 2008; Pejcha & Heyrovský 2009). With the inclusion of the space data, two additional parameters associated with the space parallax are fit: ${{\boldsymbol{\pi }}}_{{\rm{E}}}=({\pi }_{{\rm{E}},{\rm{N}}},{\pi }_{{\rm{E}},{\rm{E}}})$. We modeled the limb darkening of the source star using the parameters derived in Section 4.1. The final best-fit solutions are compiled in Table 2. The parameter errors presented are 16% and 84% confidence intervals (CIs), evaluated from the MCMC posteriors.

Table 2.  Posterior and Best-fit Microlensing Parameters Combining Ground and Space Observations

Model	${\chi }_{\mathrm{total}}^{2}$	t0 − 6817	u0	tE	$\mathrm{log}(s)$	$\mathrm{log}(q)$	α	ρ	πE,N	πE,E	fs,OGLE	fb,OGLE
		(HJD')		(days)			(rad)
u0 > 0 Median	...	0.5469	0.0039	6.454	0.2782	−0.103	−4.183	0.0240	0.0079	0.0480	5.05	0.34
68% CI (Upper)	...	0.0018	0.0007	0.032	0.0012	0.005	0.004	0.0002	0.0023	0.0008	0.04	0.04
68% CI (Lower)	...	−0.0018	−0.0007	−0.032	−0.0012	−0.005	−0.004	−0.0002	−0.0030	−0.0007	−0.04	−0.04
Best-fit	6847.596	0.5469	0.0038	6.482	0.2787	−0.105	−4.182	0.0239	0.0091	0.0477	5.02	0.37
u0 < 0 Median	...	0.5474	−0.0039	6.455	0.2782	−0.104	4.183	0.0240	0.0038	0.0484	5.05	0.34
68% CI (Upper)	...	0.0018	0.0007	0.032	0.0011	0.005	0.004	0.0002	0.0030	0.0006	0.04	0.03
68% CI (Lower)	...	−0.0018	−0.0007	−0.031	−0.0011	−0.005	−0.004	−0.0002	−0.0023	−0.0006	−0.04	−0.04
Best-fit	6847.872	0.5469	−0.0039	6.465	0.2782	−0.106	4.182	0.0239	0.0029	0.0482	5.03	0.36

Note.

DoF = 7570−11.

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Single-lens satellite parallax suffers from the well-known four-fold degeneracy (e.g., Refsdal 1966; Gould 1994). For binary lensing, depending on the data quality and coverage, some degeneracies can be resolved (see discussion in, e.g., Zhu et al. 2015). The light curve of OB140962 is well covered from both the ground and space, leading to a straightforward interpretation. In this case, the u₀ > 0 and u₀ < 0 degeneracy persists but maps into nearly identical physical properties. Therefore, the physical solution is unique. In Figure 1 we plot the u₀ > 0 solution based on the corresponding best-fitting parameters. Figure 3 shows the associated caustic structure.

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A comparison between the final fitted microlensing parameters and those from the heuristic assessment of Section 3.2 reveals that they are qualitatively consistent.

We also fit for higher-order effects in the OGLE I-band light curve, but we found that no meaningful constraints could be placed on the annual parallax and orbital motion from these data alone. This is unsurprising because these phenomena typically manifest on timescales of tens to hundreds of days. In contrast, OB140962 is magnified for merely ∼10 days, with only ∼2 days between caustic crossings.

The source star properties can be inferred from its position on the local CMD. We calculate the apparent source (V–I) color by fitting a line to the observed event flux in the OGLE V versus OGLE I bands. This yields a model-independent color of (V–I) = 1.963 ± 0.006. Modeling the source flux yields an apparent I-band magnitude of 16.22 ± 0.01.

Using the observed (V–I) versus I CMD from OGLE in the field of the target, we locate the red clump. The apparent clump centroid is (V–I, I)_clump = (2.215, 15.59). According to Bensby et al. (2013) and Nataf et al. (2013), the intrinsic color and magnitude of the red clump in the event direction are (1.06, 14.36). Using the offset between the observed and actual clump centroid as a measure of source extinction and reddening (Yoo et al. 2004), we determine the intrinsic source color and brightness to be (V–I, I)_0,source = (0.808, 14.98). This is at the blue edge of the giant clump, which is sparsely populated. From Bessell & Brett (1988)'s color table for giants, we infer the source to be a G0 giant.

We interpolate the color tables in Bessell & Brett (1988) to arrive at (V–K, K)_source = (1.78 ± 0.04, 14.00 ± 0.06). Using the relationship between stellar angular size, (V–K) color, and K magnitude given by Adams et al. (2018), we find the source angular radius θ_⋆ = 3.4 ± 0.2 μas.

The source star's brightness profile, which is important for modeling the caustic crossings, is parameterized by its limb-darkening properties. Claret & Bloemen (2011) give linear limb-darkening coefficients (u) for stars with a variety of effective temperatures, surface gravities, metallicities, and microturbulences. Assuming solar metallicity, we determine T_eff to be 5400 ± 100 K using its relation with V–I color from Casagrande et al. (2010). We adopt $\mathrm{log}(g)=3$ and microturbulence ∼2 km s⁻¹. The corresponding Γ = (2u/(3 − u)) is 0.41 in the I band and 0.14 in the Spitzer L band. These Γ values are in turn used in the final fits of the light curve parameters, as described in Section 3.3.

From the equations listed in Section 3.1, the physical properties of the system are straightforwardly determined. We display the results in Table 3. The uncertainties are derived from direct propagation. We calculate D_L, a_proj, and D_LS, assuming zero uncertainty in D_S. These final calculated properties are broadly compatible with the estimates from the earlier heuristic arguments.

Table 3.  Measured Physical Properties of Binary Lens System OGLE-2014-BLG-0962

Quantity	Value from Best-fit Solution
	u0 > 0	u0 < 0
πE	0.049 ± 0.001	0.048 ± 0.001
θE (mas)	0.144 ± 0.008	0.144 ± 0.008
μrel (mas yr−1)	8.119 ± 0.001	8.132 ± 0.001
Mtot (M☉)	0.365 ± 0.020	0.366 ± 0.020
${M}_{\mathrm{prim}}\ ({M}_{\odot })$	0.204 ± 0.011	0.205 ± 0.011
Msec (M☉)	0.160 ± 0.009	0.161 ± 0.009
DS (kpc)	7.863 ± 0.000	7.863 ± 0.000
DL (kpc)	7.453 ± 0.021	7.455 ± 0.021
aproj (au)	2.040 ± 0.107	2.036 ± 0.107
D8.3 (kpc)	7.845 ± 0.021	7.847 ± 0.021
DLS	0.410 ± 0.021	0.407 ± 0.021

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Since the exact source distance is unknown, we list both D_8.3 (Calchi Novati et al. 2015a; see also Section 5.4) and D_L relative to the mean bar clump distance at the event's Galactic coordinates, which is D_S = 7.86 kpc (Nataf et al. 2013). Regardless of the precise location of the source, this M+M-dwarf binary is the most distant lensing system discovered with Spitzer.

The Bayesian analysis framework is based on a Galactic model prior, whose ingredients are mass functions (MF) and both velocity distributions (VD) and density profiles (DP) of the bulge and disk of the Milky Way, in the direction of the event. Each draw of a lens–source pair has a corresponding θ_E and μ_rel (which is interchangeable with t_E). For binary lenses, we make the following modification to the original Bayesian formalism, whose MF assumes single stars and does not account for binaries. We draw the mass of the primary component, M_prim, from an appropriate MF. Then we calculate M_tot and θ_E from M_tot = M_prim (1 + q). One underlying assumption is that the binary parameters of the event (s, q) do not depend on the mass and distance of the lens. The likelihood function is constrained by the observed t_E and θ_E of the event.

Our baseline Galactic model draws from the single-star MF reported by Chabrier (2003) and derives the VD and DP from Han & Gould (1995a, 2003, hereafter collectively denoted HG). Specifically, the disk DP follows the exponential functional form of Bahcall (1986), with scale heights and thin/thick disk density normalization constants from Jurić et al. (2008). The bulge DP parameters are based on the barred model ("G2") from Dwek et al. (1995), scaled to the stellar mass density given in Batista et al. (2011). This is described in detail in Jung et al. (2018).

We choose to test two VDs and DPs: the baseline model (HG) and that used by Zhu et al. (2017, hereafter Zhu17). The Zhu17 bulge DP and its normalization are based on Robin et al. (2003), which is more compact and has a higher stellar density contrast relative to its disk (Han & Gould 1995a) than that for our baseline model. In addition, the bulge velocity dispersions are also markedly larger for the Zhu17 VD (σ_y,z,B = 120 km s⁻¹) than for HG (σ_y,B = 82.5 km s⁻¹, σ_z,B = 66.3 km s⁻¹). Table 4 shows the four model combinations investigated in this work. We note that the Chabrier and Sumi (Sumi et al. 2011) MFs are very similar, and it has been previously demonstrated that the choice of MF makes little difference in inferring lens parameters, at least for single lenses (e.g., Zhu et al. 2017). Therefore, we use the Chabrier MF for all cases tested, excluding stellar remnants for events involving planetary lenses.

We input our best-fit θ_E and t_E values from the u₀ > 0 solution into a Bayesian analysis with our baseline Galactic model. Figure 4 displays the posterior distributions for M_prim and D_LS. For lenses in the bulge, D_LS is a more robust metric than D_L, since it is comparatively less sensitive to the uncertainty in the source distance. The directly measured quantities are overplotted. Bayesian analysis indicates that the lens's primary has $\mathrm{log}({M}_{\mathrm{prim}}/{M}_{\odot })=-{1.17}_{-0.35}^{+0.43}$ or ${M}_{\mathrm{prim}}={0.07}_{-0.04}^{+0.11}\,{M}_{\odot }$. The true value of M_prim ∼ 0.20 is just outside the 68% CI. In this case, the difference between the Bayesian value and the true value means the difference between a BD binary detection and a run-of-the-mill M+M binary. Similarly, taking D_S to be the clump distance in the direction of this event (Nataf et al. 2013), we find that the parallax-derived value of D_LS = 0.41 kpc is just outside the 68% CI of the Bayesian prediction for the lens–source distance ${D}_{{LS}}={1.31}_{-0.77}^{+1.22}$ kpc. We repeat this analysis for Galactic models 2–4. In case 2, ${M}_{\mathrm{prim}}={0.08}_{-0.04}^{+0.11}\,{M}_{\odot }$ and ${D}_{{LS}}={1.11}_{-0.61}^{+0.83}$ kpc. Case 3 returns ${M}_{\mathrm{prim}}={0.08}_{-0.04}^{+0.13}\,{M}_{\odot }$ and ${D}_{{LS}}={1.28}_{-0.75}^{+1.16}$ kpc. In case 4, ${M}_{\mathrm{prim}}={0.11}_{-0.05}^{+0.16}\,{M}_{\odot }$ and ${D}_{{LS}}\,={0.92}_{-0.51}^{+0.78}$ kpc.

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The very small π_E measured by Spitzer places the true lens very close to the source star. In all cases, the mass and distance (D_L = D_S − D_LS) are underpredicted. It is clear that Spitzer has provided important added value for accurately determining the mass and distance to this microlens.

Another noteworthy lens with Spitzer parallax is OB161195 (Shvartzvald et al. 2017), which has been independently analyzed by Bond et al. (2017) using Bayesian analysis alone. The physical properties derived from parallax are ${M}_{L}={0.078}_{-0.012}^{+0.016}\,{M}_{\odot }$ and ${D}_{L}={3.91}_{-0.46}^{+0.42}$ kpc (Shvartzvald et al. 2017). This is between the 68% and 95% CIs of Bond et al. (2017), which yielded ${M}_{L}={0.37}_{-0.21}^{+0.38}$ and ${D}_{L}\,={7.20}_{-1.02}^{+0.85}$.

Our Bayesian posteriors for this lens system according to the baseline model are ${M}_{L}={0.20}_{-0.11}^{+0.25}\,{M}_{\odot }$ and ${D}_{L}={6.41}_{-1.49}^{+1.20}$ kpc, which differ from those of Bond et al. (2017). The main source of the discrepancy arises from differences in the Galactic models assumed. Bond et al. (2017) uses a bulge VD with σ_y = 103.8 km s⁻¹, σ_z = 96.4 km s⁻¹, but also includes a bulge (solid body) rotation of 50 km s⁻¹ kpc⁻¹. In addition, the VDs are truncated at 600 km s⁻¹ in the bulge and 550 km s⁻¹ in the other components. For their DP, they also include thick disk and spheroid components. They use the Sumi MF, but this should not have a significant effect because of the similarity to the Chabrier MF. Finally, we note that our analysis is based on the t_E and θ_E given in Shvartzvald et al. (2017), whereas Bond et al. (2017) measure a slightly larger t_E and smaller θ_E. This does have a small effect on the results, but the biggest effects are due to differences in the Galactic models.

The case of OB161195 indicates that the assumption of Galactic models can affect the conclusions. At the same time, it is worth noting that OB161195 is kinematically peculiar: despite its disk-like distance, its motion is not in the direction of the disk's rotation (Shvartzvald et al. 2017). For a Bayesian analysis based on θ_E and t_E, the direction of the motion would be unknown. Therefore, with this knowledge, we recognize that a Bayesian analysis may not accurately reflect the properties of OB161195.

Bayesian analysis is expected to give a statistical representation of the truth. As such, it is not in itself surprising that individual outliers like OB140962 and OB161195 exist. The growing inventory of objects with Spitzer satellite parallaxes allows us to investigate whether there are systematic problems with the Bayesian framework for a larger sample, specifically the consistency with and dependence on Galactic models.

Here we describe a set of 13 published Spitzer events (including OB140962) with unique measurements of both π_E and θ_E, for which we test the Bayesian analyses. Their relevant measured properties are given in Tables 5 and 6, where the tabulated lens masses, distances, and their uncertainties are computed from the θ_E and π_E values given in the literature using Equation (4). Among their ranks are both two-body lenses and single lenses with securely modeled finite-source effects. For events with degeneracies for which the authors advocate strongly for one particular solution for θ_E and π_E based either on χ² fitness or on physical grounds (OB140289: Udalski et al. 2018; OB141050: Zhu et al. 2015; OB161045: Shin et al. 2018; OB161190: Ryu et al. 2018), we retain the favored solutions only. Events with degeneracies yielding physical properties consistent within 1σ are assigned physical properties corresponding to the solution with the lowest χ² for this comparison (OB140124: Udalski et al. 2015b; Beaulieu et al. 2018; OB141050: Zhu et al. 2015; OB150479: Han et al. 2016; OB161195: Shvartzvald et al. 2017). Note that, although membership in our sample requires selection for Spitzer follow-up, it is not necessarily true that both θ_E and π_E were immediately constrained by the Spitzer plus ground observations. For OB140124, θ_E is not well measured, due to the absence of the finite source effect. Therefore, we assign to it physical parameters determined by follow-up AO imaging (Beaulieu et al. 2018). In the case of OB140289, Spitzer could not constrain π_E because it happened to observe a featureless region of the light curve. Fortuitously, this event is sufficiently long that annual parallax could be accurately and precisely determined.

Table 5.  Spitzer Bayesian and Binary Sample Properties

Object	Deg	l	b	θE	tE (days)	πE	${\mu }_{\mathrm{hel},l}$	μhel,b	D8.3	ML,tot	qa	Bayesb	Binc	Referencesd
Abbrev	Solution						(mas yr−1)	(mas yr−1)	(kpc)	(M☉)
OB140124	++	2.34	−2.92	1.030 ± 0.060	150.80 ± 2.80	0.146	1.60	−3.02	3.7	0.90	6.9e−04	Y	Y	(1), (15)
OB140289	++	0.80	−1.62	1.170 ± 0.090	144.43 ± 0.24	0.152	1.40	1.91	3.4	0.94	8.1e−01	Y	Y	(2)
OB140962		2.66	−2.54	0.144 ± 0.008	6.45 ± 0.03	0.049	5.12	−6.36	7.8	0.36	7.9e−01	Y	Y	(3)
OB141050	++	5.09	3.23	1.340 ± 0.160	73.30 ± 4.20	0.120	1.48	−7.38	3.6	1.37	3.9e−01	Y	Y	(4)
OB150020		−2.24	−3.16	1.329 ± 0.049	63.55 ± 0.06	0.223	−5.88	−4.67	2.4	0.73	2.1e−01	Y	Y	(5)
OB150479	−	−5.82	−3.08	1.870 ± 0.430	86.30 ± 0.50	0.125	−6.09	2.86	2.8	1.83	8.1e−01	Y	Y	(6)
OB150763	−+	−1.85	2.25	0.288 ± 0.020	32.78 ± 0.25	0.071	2.94	1.32	7.1	0.50	⋯	Y	N	(7)
OB150966	−close	0.96	−1.82	0.760 ± 0.070	57.80 ± 0.40	0.241	−2.57	2.70	3.3	0.39	1.7e−04	Y	Y	(8)
OB151268		7.34	1.42	0.127 ± 0.009	17.50 ± 0.70	0.347	2.68	−0.97	6.1	0.05	⋯	Y	N	(7)
OB151319	−+ wide	−1.71	−4.05	0.660 ± 0.070	98.80 ± 4.80	0.124	−0.02	2.00	4.9	0.65	9.5e−02	N	Y	(9)
OB160168		−1.84	−2.42	1.410 ± 0.120	93.67 ± 1.17	0.363	6.12	1.13	1.6	0.48	7.7e−01	Y	Y	(10)
OB161045	−+	−5.75	−1.39	0.245 ± 0.015	11.98 ± 0.08	0.355	5.86	−5.41	4.8	0.08	⋯	Y	N	(11)
OB161190		2.62	−1.84	0.490 ± 0.040	93.53 ± 0.89	0.067	1.77	0.67	6.5	0.90	1.5e−02	Y	Y	(12)
OB161195	−+ close	−0.00	−2.48	0.286 ± 0.050	9.96 ± 0.11	0.473	0.29	9.76	3.9	0.07	5.5e−05	Y	Y	(13)
OB161266	A–	−0.04	−1.50	0.227 ± 0.011	8.65 ± 0.08	0.971	9.92	−0.17	2.9	0.03	7.6e−01	N	Y	(14)

Notes.

^a⋯ denotes single lenses for which the notion of q is not applicable. ^bWhether or not the object is included in the Bayesian analysis. ^cWhether or not the object is included in the binarity analysis. ^dReferences. (1) Udalski et al. (2015b), (2) Udalski et al. (2018), (3) this work, (4) Zhu et al. (2015), (5) Wang et al. (2017), (6) Han et al. (2016), (7) Zhu et al. (2016), (8) Street et al. (2016), (9) Shvartzvald et al. (2016a), (10) Shin et al. (2017), (11) Shin et al. (2018), (12) Ryu et al. (2018), (13) Shvartzvald et al. (2017), (14) Albrow et al. (2018), (15) Beaulieu et al. (2018).

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Table 6.  Lens Properties: Measured vs. Bayesian

Object	Measured		Bayesian Model 1		Bayesian Model 2		Bayesian Model 3		Bayesian Model 4
Abbrev	Mprim (M☉)	D8.3 (kpc)	Mprim (M☉)	D8.3 (kpc)	Mprim (M☉)	D8.3 (kpc)	Mprim (M☉)	D8.3 (kpc)	Mprim (M☉)	D8.3 (kpc)
OB140124a	0.90 ± 0.05	3.70 ± 0.20	${0.78}_{-0.36}^{+0.41}$	${3.45}_{-1.13}^{+0.90}$	${0.72}_{-0.33}^{+0.41}$	${3.29}_{-1.11}^{+0.92}$	${0.82}_{-0.38}^{+0.42}$	${3.53}_{-1.17}^{+0.90}$	${0.75}_{-0.35}^{+0.40}$	${3.36}_{-1.13}^{+0.96}$
OB140289b	0.52 ± 0.04	3.35 ± 0.16	${0.75}_{-0.32}^{+0.38}$	${4.06}_{-1.12}^{+0.95}$	${0.69}_{-0.31}^{+0.33}$	${3.82}_{-1.07}^{+0.90}$	${0.78}_{-0.32}^{+0.55}$	${4.19}_{-1.14}^{+1.11}$	${0.72}_{-0.34}^{+0.45}$	${3.94}_{-1.19}^{+1.14}$
OB140962	0.20 ± 0.01	7.85 ± 0.02	${0.07}_{-0.04}^{+0.11}$	${7.11}_{-1.10}^{+0.74}$	${0.08}_{-0.04}^{+0.11}$	${7.32}_{-0.80}^{+0.57}$	${0.08}_{-0.04}^{+0.13}$	${7.21}_{-0.95}^{+0.69}$	${0.11}_{-0.05}^{+0.16}$	${7.50}_{-0.68}^{+0.50}$
OB141050	0.99 ± 0.28	3.55 ± 0.57	${0.82}_{-0.33}^{+0.54}$	${3.36}_{-1.04}^{+1.09}$	${0.80}_{-0.33}^{+0.56}$	${3.27}_{-1.04}^{+1.13}$	${0.82}_{-0.33}^{+0.54}$	${3.28}_{-1.01}^{+1.12}$	${0.80}_{-0.33}^{+0.58}$	${3.20}_{-1.01}^{+1.20}$
OB150020	0.60 ± 0.03	2.40 ± 0.07	${0.88}_{-0.38}^{+0.55}$	${3.11}_{-1.02}^{+1.08}$	${0.86}_{-0.38}^{+0.56}$	${3.06}_{-1.05}^{+1.11}$	${0.89}_{-0.34}^{+0.51}$	${3.09}_{-0.88}^{+0.91}$	${0.86}_{-0.32}^{+0.52}$	${3.03}_{-0.87}^{+0.96}$
OB150479	1.01 ± 0.25	2.82 ± 0.45	${0.82}_{-0.33}^{+0.55}$	${3.49}_{-1.35}^{+1.41}$	${0.79}_{-0.33}^{+0.54}$	${3.41}_{-1.35}^{+1.41}$	${0.86}_{-0.35}^{+0.57}$	${3.34}_{-1.24}^{+1.43}$	${0.84}_{-0.35}^{+0.58}$	${3.26}_{-1.23}^{+1.44}$
OB150763	0.50 ± 0.03	7.09 ± 0.07	${0.42}_{-0.21}^{+0.31}$	${6.90}_{-0.99}^{+0.56}$	${0.57}_{-0.23}^{+0.26}$	${7.24}_{-0.56}^{+0.33}$	${0.42}_{-0.22}^{+0.30}$	${6.90}_{-1.08}^{+0.56}$	${0.57}_{-0.23}^{+0.27}$	${7.24}_{-0.57}^{+0.34}$
OB150966	0.39 ± 0.04	3.30 ± 0.19	${0.76}_{-0.37}^{+0.41}$	${4.70}_{-1.39}^{+0.92}$	${0.69}_{-0.35}^{+0.44}$	${4.46}_{-1.40}^{+1.09}$	${0.73}_{-0.36}^{+0.40}$	${4.57}_{-1.38}^{+0.98}$	${0.65}_{-0.33}^{+0.44}$	${4.31}_{-1.38}^{+1.13}$
OB151268	0.05 ± 0.01	6.08 ± 0.35	${0.11}_{-0.06}^{+0.16}$	${7.19}_{-0.93}^{+0.67}$	${0.14}_{-0.07}^{+0.20}$	${7.47}_{-0.63}^{+0.50}$	${0.11}_{-0.06}^{+0.16}$	${7.17}_{-1.00}^{+0.68}$	${0.14}_{-0.07}^{+0.20}$	${7.47}_{-0.67}^{+0.48}$
OB160168	0.27 ± 0.03	1.58 ± 0.14	${0.87}_{-0.34}^{+0.53}$	${3.65}_{-1.01}^{+1.07}$	${0.85}_{-0.34}^{+0.56}$	${3.58}_{-1.02}^{+1.13}$	${0.85}_{-0.34}^{+0.54}$	${3.58}_{-1.02}^{+1.03}$	${0.84}_{-0.35}^{+0.60}$	${3.54}_{-1.05}^{+1.19}$
OB161045	0.08 ± 0.01	4.82 ± 0.15	${0.24}_{-0.13}^{+0.33}$	${6.67}_{-1.34}^{+0.87}$	${0.30}_{-0.18}^{+0.32}$	${6.94}_{-1.44}^{+0.67}$	${0.28}_{-0.16}^{+0.32}$	${6.85}_{-1.25}^{+0.72}$	${0.44}_{-0.25}^{+0.28}$	${7.30}_{-0.99}^{+0.39}$
OB161190	0.88 ± 0.08	6.52 ± 0.13	${0.67}_{-0.33}^{+0.27}$	${6.12}_{-1.32}^{+0.51}$	${0.81}_{-0.33}^{+0.18}$	${6.44}_{-0.90}^{+0.33}$	${0.60}_{-0.32}^{+0.32}$	${5.92}_{-1.50}^{+0.68}$	${0.74}_{-0.44}^{+0.23}$	${6.31}_{-1.68}^{+0.43}$
OB161195	0.07 ± 0.01	3.91 ± 0.38	${0.20}_{-0.11}^{+0.25}$	${6.41}_{-1.33}^{+0.95}$	${0.26}_{-0.14}^{+0.26}$	${6.91}_{-1.23}^{+0.66}$	${0.27}_{-0.14}^{+0.27}$	${6.58}_{-1.13}^{+0.78}$	${0.38}_{-0.19}^{+0.29}$	${7.05}_{-0.80}^{+0.51}$
p-Value	⋯	⋯	0.42	0.17	0.53	0.34	0.24	0.17	0.29	0.30

Notes. Measured values are calculated from θ_E and π_E measured based on Spitzer parallax as reported in the literature, except where noted otherwise; Bayesian values shown are median and symmetric 68% CIs.

^aMeasured values come from the Spitzer parallax plus AO imaging (Beaulieu et al. 2018). ^bMeasured values come from the finite source effect plus precise ground-based modeling of the annual parallax (Udalski et al. 2018).

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We exclude from our Bayesian sample the events with severe degeneracies, that is, those with multiple solutions for which the physical properties M_L and D_L are incompatible within their nominal uncertainties (OB150196: Han et al. 2017; OB151482: Chung et al. 2017; OB170329: Han et al. 2018). It would be difficult to interpret a comparison between the Bayesian results and quantities that are ill defined. We also exclude OB151285 (Shvartzvald et al. 2015) and OB161266 (Albrow et al. 2018) from this exercise because the Bayesian priors for the MF of planetary-mass objects and stellar remnants are not well understood.

For the purpose of this comparison, we use

$$\begin{eqnarray}&&{D}_{8.3}=\displaystyle \frac{\mathrm{kpc}}{{\pi }_{\mathrm{rel}}/\mathrm{mas}+1/8.3}\end{eqnarray} \tag{ 7 }$$

as our distance metric (Calchi Novati et al. 2015a). This metric is independent of uncertainties in the source distance and is well defined for lenses in both the disk and the bulge. The first point is significant, because the literature does not homogeneously report the source distance assumed in the D_L calculation, and the Bayesian analysis draws sources from a range of distances in the bulge.

As for OB140962 (Section 5.2), we now compute Bayesian posteriors for lens mass and distance for all 13 objects using galactic models given in Table 4. We treat binaries as described in Section 5.1. For planetary events (defined for this purpose to be q < 0.05), we exclude stellar remnants from the Galactic models. The results are summarized in Table 6.

Figure 5 shows the comparison between the output from the baseline Galactic model and the values calculated from parallax. For the majority of the cases, the Bayesian posteriors are consistent with the true measured values. OB140962 is one of the objects for which the true values are in tension with the Bayesian predictions (see Section 5.2). For both mass and distance, the true values lie above the 84th percentile of the Bayesian posteriors. However, if the posteriors are true representations of the data, occasional outliers are expected. In fact, this particular situation should occur for 16% of the instances.

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To test the overall accuracy of the baseline model, we evaluate the fraction of posterior lying above the true values of M_L and D_8.3 derived from parallax for each object in this ensemble. If these Bayesian posteriors represent the true values fairly for this ensemble, we should expect this cumulative distribution function (CDF) to follow the identity function (e.g., 10% of the time the true value falls below 10% of the posterior). In Figure 6, the black solid line shows the cumulative distribution of the fraction of posterior lying above the parallax value for M_L and D_8.3. One-sample Kolmogorov–Smirnov (KS) tests show that the distributions for the two parameters are consistently distributed around the one-to-one line. This indicates that, on average, Bayesian analysis with our baseline Galactic model is a reasonable reflection of the data. Of course, a direct measurement (e.g., with parallax or AO observations) is still necessary to correctly identify the nature of any given individual object.

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Agreement with the other three Galactic models tested are comparably good. The last row of Table 6 shows the p values from the one-sample KS test for the Bayesian posteriors of the two physical parameters for each model, with their CDFs also plotted in Figure 6. The KS test shows that all models are overall consistent with the Spitzer sample. However, there are some systematic effects between model predictions. For objects perceived to be near the Galactic bulge (Bayes D_8.3 ≳ 6 kpc), models using the Zhu17 DPs tend to predict greater lens mass and distance than those from the HG DPs (see Table 6). This can be attributed to the relatively higher stellar density and compactness of the bulge component in the Zhu17 DP compared to in HG. Therefore, the DP of Zhu17 has a greater tendency to "pull" a lens that already appears to be close to the bulge even closer in toward the Galactic center, which increases the inferred distance. To conserve the observed θ_E, the inferred mass is increased as well (see Equation (2)). While this discrepancy is much smaller than the typical uncertainties, it is a systematic effect that should be acknowledged when combining lens properties inferred from Bayesian analyses with different Galactic model priors.